Question 3.5.8: Find the sum and the difference of the identity function and......

Let f: R \rightarrow R: f(x)=x be the identity function.

And, let g: R \rightarrow R: g(x)=|x| be the modulus function.

Then, \operatorname{dom}(f)=R and \operatorname{dom}(g)=R .

\therefore \quad \operatorname{dom}(f) \cap \operatorname{dom}(g)=R \cap R=R .

(i) \operatorname{dom}(f+g)=\operatorname{dom}(f) \cap \operatorname{dom}(g)=R .

Now, (f+g): R \rightarrow R is given by

Hence, (f+g)(x)=\left\{\begin{array}{l}2 x, \text { when } x \geq 0 \\ 0, \text { when } x<0 .\end{array}\right.

(ii) \operatorname{dom}(f-g)=\operatorname{dom}(f) \cap \operatorname{dom}(g)=R .